Calculation of the derivative of interpolation shape function for three-dimensional natural element method

Abstract

The natural element method (NEM) is a special meshless method. Its shape functions are constructed using natural neighbor node interpolations based on the concepts of Voronoi tessellation. The NEM interpolation is linear between adjacent nodes on the boundary of the convex hull, which facilitates imposition of essential boundary conditions. However, for a three-dimensional problem, the computation of shape function derivative of NEM is still very complicated even with the non-Sibson interpolation function, which makes the NEM an unpopular numerical method. In this paper, we adopt the direct mathematical derivative technique, and after some rigorous deduction, finally obtain the shape function derivative expression of three-dimensional NEM. Compared with the Lasserre algorithm, this algorithm is more intuitionistic and can be conveniently programmed. The NEM numerical results for cantilever beams verify the correctness of the shape function derivative expression of NEM derived in this paper.